Class Of R-modules Research Articles

On (co)pure Baer injective modules

For a given class of R-modules Q, a module M is called Q-copure Baer injective if any map from a Q-copure left ideal of R into M can be extended to a map from R into M. Depending on the class Q, this concept is both a dualization and a generalization of pure Baer injectivity. We show that every module can be embedded as Q-copure submodule of a Q-copure Baer injective module. Certain types of rings are characterized using properties of Q-copure Baer injective modules. For example a ring R is Q-coregular if and only if every Q-copure Baer injective R-module is injective.

Lower bounds of certain general local cohomology modules

Let R be a commutative Noetherian ring, a system of ideals of R, M an arbitrary R-module and t a non-negative integer. Let be a Melkersson subcategory of R-modules. Among other things, we prove that if is in for all i < t then is in for all i < t and for all If is the class of R-modules N with where is an integer, then is in for all i < t if (and only if) is in for all i < t and for all As consequences we study and compare vanishing, Artinianness and support of general local cohomology and ordinary local cohomology supported at ideals of its system of ideals at initial points i < t. We show that is not necessarily finite whenever is local and M a finitely generated R-module.Communicated by Ellen Kirkman

From Hom(A,X)  Hom(B,X) to A  B

Let A and B be two R-modules. We examine conditions under which Hom(A,X) ? Hom(B,X), implies that A ? B, where X belongs to an appropriate class of R-modules. Different perspectives of the question are studied. In the case of abelian groups (Z-modules), this investigation gives a partial answer to an old problem of L. Fuchs.

Projective and injective dimensions in the category σ[M

ABSTRACTFor any module M, the smallest subcategory of R-modules which is a Grothendieck category and contains M is the category σ[M]. In this article, by introducing two classes of R-modules under the titles -modules and Σn-modules, the notions σ-projective and σ-injective dimension in the category σ[M] have been defined and studied. The relationships between these notions and the corresponding notions defined in the category of R-modules have been investigated. Also, σ-global dimension is defined and some of its properties have been studied.

Cartan-Eilenberg projective, injective and flat complexes

Let R be an associative ring with identity and F a class of R-modules. In this article: we first give a detailed treatment of Cartan-Eilenberg F complexes and extend the basic properties of the class F to the class CE(F). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these complexes, including coherent, Noetherian, von Neumann regular rings, QF rings, semisimple rings, hereditary rings and perfect rings.

ON REFLEXIVE MODULES OVER COMMUTATIVE RINGS

Abstract. Let R be a commutative ring and U an R-module. The aimof this paper is to study the duality between U-reflexive (pre)envelopesand U-reflexive (pre)covers of R-modules. 1. Introduction and preliminariesLet Rbe a commutative ring and Uan R-module. For R-modules MandN, Hom(M,N) means Hom R (M,N), and Ext n (M,N) means Ext nR (M,N) foran integer n≥ 1. For an R-module M, Hom(M,U) is called the dual moduleof Mwith respect to Uand denoted by M ∗ . For a homomorphism fbetweenR-modules, we put f ∗ = Hom(f,U). Let δ M : M→ M ∗∗ via δ M (x)(f) = f(x)for any x∈ Mand f∈ M ∗ be the canonical evaluation homomorphism. If δ M is an isomorphism, then Mis called a U-reflexive module. We denote by R U the class of U-reflexive modules.Let Rbe a Noetherian ring. A finitely generated R-module Mis said to haveGorensteindimension (abbr. G-dimension) zero[2]if M∼=Hom(Hom(M,R),R)(i.e., M is R-reflexive); and Ext i (M,R) = 0 = Ext i (Hom(M,R),R) for alli≥ 1. We denote by G(R) the class of R-modules having G-dimension zero.Let F be a class of R-modules and M an R-module. A homomorphismφ : M → F with F ∈ F is called an F-preenvelope of M [10] if for anyhomomorphism f: M→ F

On the Regularity and Koszulness of Modules Over Local Rings

Koszul modules over Noetherian local rings R were introduced by Herzog and Iyengar and they possess good homological properties, for instance their Poincaré series is rational. It is an interesting problem to characterize classes of Koszul modules. Following the idea traced by Avramov, Iyengar, and Sega, we take advantage of the existence of special filtration on R for proving that large classes of R-modules over Koszul rings are Koszul modules. By using this tool we reprove and extend some results obtained by Fitzgerald.

Stability of Gorenstein Classes of Modules

The purpose of this paper is to give, via totally different techniques, an alternate proof to the main theorem of [18] in the category of modules over an arbitrary ring R. In effect, we prove that this theorem follows from establishing a sequence of equalities between specific classes of R-modules. Actually, we tackle the following natural question: What notion emerges when iterating the very process applied to build the Gorenstein projective and Gorenstein injective modules from complete resolutions? In other words, given an exact sequence of Gorenstein injective R-modules G= ⋯ → G1→ G0→ G-1→ ⋯ such that the complex Hom R(H,G) is exact for each Gorenstein injective R-module H, is the module Im (G0→ G-1) Gorenstein injective? We settle such a question in the affirmative and the dual result for the Gorenstein projective modules follows easily via a similar treatment to that used in this paper. As an application, we provide the Gorenstein versions of the change of rings theorems for injective modules over an arbitrary ring.

Some Results on the Finiteness of Generalized Local Cohomology Modules Using Serre Classes

Let R be a commutative Noetherian ring with identity and \(\mathfrak{a}\) be an ideal of R. In this paper, we investigate the conditions under which the set of associated primes of generalized local cohomology modules is finite. For this aim, we study some relations between generalized local cohomology modules, ideal transform, Ext and ordinary local cohomology modules in the Serre classes of R-modules. Such relations provided a common language for expressing some results about the generalized local cohomology R-modules, that has appeared in some papers.

Preservation of quasi-isomorphisms of complexes

We consider the preservation property of the homomorphism and tensor product functors for quasi-isomorphisms and equivalences of complexes. Let X and Y be two classes of R-modules with Ext〉-1(X, Y ) = 0 for each object X ∈ X and each object Y ∈ Y. We show that if A,B ∈ C ⊐(R) are X-complexes and U, V ∈ C ⊏(R) are Y-complexes, then Open image in new window ; Open image in new window . As an application, we give a sufficient condition for the Hom evaluation morphism being invertible.

Gorenstein projective and flat complexes over noetherian rings

AbstractWe give sufficient conditions on a class of R‐modules \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}$\end{document} in order for the class of complexes of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}$\end{document}‐modules, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$dw \mathcal {C}$\end{document}, to be covering in the category of complexes of R‐modules. More precisely, we prove that if \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}$\end{document} is precovering in R − Mod and if \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}$\end{document} is closed under direct limits, direct products, and extensions, then the class \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$dw \mathcal {C}$\end{document} is covering in Ch(R). Our first application concerns the class of Gorenstein flat modules. We show that when the ring R is two sided noetherian, a complex C is Gorenstein flat if and only if each module Cn is Gorenstein flat. If moreover every direct product of Gorenstein flat modules is a Gorenstein flat module, then the class of Gorenstein flat complexes is covering. We consider Gorenstein projective complexes as well. We prove that if R is a commutative noetherian ring of finite Krull dimension, then the class of Gorenstein projective complexes coincides with that of complexes of Gorenstein projective modules. We also show that if R is commutative noetherian with a dualizing complex then every right bounded complex has a Gorenstein projective precover.

Covers and Envelopes by #-ℱ Complexes

Let ℱ be a class of R-modules. #-ℱ complexes are introduced and investigated in this article. We say that a complex C is a #-ℱ complex if all terms C i are in ℱ for i ∈ ℤ, and we denote the class of #-ℱ complexes by . Some characterizations of #-ℱ complexes are given. The existence of -(pre)envelopes and -(pre)covers of complexes is discussed, and as a consequence, some known results are recovered. Some relations between -(pre)covers of a complex X and ℱ-(pre)covers of the R-modules X i are also studied.

Covers and envelopes with respect to a semidualizing module

Let R be a commutative ring and C be a semidualizing R-module. For a given class of R-modules Q, we define a class Q_C by M \in Q_C \Leftrightarrow Hom_R(C,M) \in Q. We prove that if Q \subseteq (R) is a Kaplansky class and closed under direct sums, then Q_C^{\bot} is special preenveloping. As corollaries, we can show that p_C^{n \bot} and f_C^{n \bot} are both special preenveloping. Finally, we show that I_C^n is covering, I_C^{n \bot} is enveloping and special preenveloping provided R is Noetherian.

Ideals, Natural Classes, and Functors

For any ring R, the set 𝒩(R) of all natural classes of R-modules is a complete Boolean lattice, which is a direct sum of two convex and complete Boolean sublattices 𝒩(R) = 𝒩 t (R) ⊕ 𝒩 f (R), where the last summand is the set of all nonsingular natural classes. The ring R contains a unique lattice of ideals 𝒥(R) which is lattice isomorphic to 𝒩 f (R). The present note develops the analogue of all of the above for an arbitrary R-module M, so that in the special case when M R = R R , the known lattice isomorphism 𝒥(R) ≅ 𝒩 f (R) is recovered.

Elementary varieties and existence of flat covers

It is shown that flat covers exist in a wide class of additive categories – we call them elementary – which behave similar to locally finitely presented Grothendieck categories. Elementary categories have enough “finitely presented” objects, but they need not be locally finitely presented. This is related to the existence of pure monomorphisms which are not kernels and the non-exactness of direct limits. For a module category Mod ( R ) , every class of R-modules containing R cogenerates an elementary full subcategory.

Jacobson Radical Isomorphism Theorems for Mixed Modules Part One: Determining the Torsion

Let R be a complete discrete valuation domain. A class of R-modules satisfies an isomorphism theorem if modules in that class are determined by their endomorphism rings. Do there exist classes of mixed R-modules for which the Jacobson radical of the endomorphism ring determines the structure of the modules? As a first step in answering this question, this article shows that there are two classes of mixed R-modules for which the Jacobson radical of the endomorphism ring determines the structure of the maximal torsion submodules of modules in that class.

Formula omitted]-superfluous and [formula omitted]-essential homomorphisms

Let R be a ring and S a class of R-modules. S -superfluous epimorphisms and S-essential monomorphisms are introduced and studied in this article. As applications, some new characterizations of von Neumann regular rings and perfect rings are given. Finally, these notions are also used to study minimal homomorphisms.

Modules Which Are Lifting Relative To Module Classes

In this paper, we study a module which is lifting and supplemented relative to a module class. Let R be a ring, and let X be a class of R-modules. We will define X-lifting modules and X-supplemented modules. Several properties of these modules are proved. We also obtain results for the case of specific classes of modules.

Butler Modules Over Prüfer Domains

If R is an integral domain, let 𝒞 be the class of torsion free completely decomposable R-modules of finite rank. Denote by ℛ the class of those torsion-free R-modules A such that A is a homomorphic image of some C ∈ 𝒞, and let 𝒫 be the class of R-modules K such that K is a pure submodule of some C ∈ 𝒞. Further, let Q ℛ and Q 𝒫 be the respective closures of ℛ and 𝒫 under quasi-isomorphism. In this article, it is shown that if R is a Prüfer domain, then Q ℛ = Q 𝒫, and ℛ = 𝒫 in the special case when R is h-local. Also, if R is an h-local Prüfer domain and if C ∈ 𝒞 has a linearly ordered typeset, it is established that all pure submodules and all torsion-free homomorphic images of C are themselves completely decomposable. Finally, as an application of these results, we prove that if R is an h-local Prüfer domain, then ℛ = Q ℛ = Q 𝒫 = 𝒫 if and only if R is almost maximal.

ℒ-Injective Hulls of Modules

Let R be a ring and ℒ a class of R-modules. An R-module N is called ℒ-injective if for all L ∈ ℒ. An ℒ-injective hull of an R-module M is defined to be a homomorphism φ: M → F with F ℒ-injective such that for any monomorphism f: M → F′ with F′ ℒ-injective, there is a monomorphism g: F → F′ satisfying gφ = f. The aim of this paper is to study ℒ-injective hulls and their relations with ℒ-injective envelopes in Enochs' sense.